Finding Nominal Rate

[reardear]

An investment of $25000 accumulated to $38725 over 20 years where interest was compounded bi-weekly. Find the nominal rate of interest (also compounded bi-weekly) as a percentage.

I've done financial questions almost a year ago but want to know if I can do this without looking back at old notes (I did have to find the formula though ). This is what I've done so far:

\(\displaystyle 38725 = 25000(1+\frac{r}{26})^{520}\)
\(\displaystyle 1.549 = (1+\frac{r}{26})^{520}\)

and then... use ln?

 

[MarkFL]

You want the exponent to reflect the number of compounding periods, that is the number of years times the number of times per years the interest is compounded.

 

[JeffM]

An investment of $25000 accumulated to $38725 over 20 years where interest was compounded bi-weekly. Find the nominal rate of interest (also compounded bi-weekly) as a percentage.

I've done financial questions almost a year ago but want to know if I can do this without looking back at old notes (I did have to find the formula though ). This is what I've done so far:

\(\displaystyle 38725 = 25000(1+\frac{r}{26})^{20}\)
\(\displaystyle 1.549 = (1+\frac{r}{26})^{20}\)

and then... use ln?

Any consistent type of log and antilog will do. Of course, as MarkFL says, you will always get the wrong answer with the wrong exponent.

 

[reardear]

Okay I thought something would be wrong . I'm not sure about using ln for the fraction inside, so could I do this?

\(\displaystyle \large 1.549^{\frac{1}{520}} = 1 + \frac{r}{26}\)
\(\displaystyle 0.00084191 = \frac{r}{26}\)
\(\displaystyle r = 0.0218897\)
\(\displaystyle r = 2.19\%\)?

Edit: Plugged it in and it came out right, so I guess so Thanks for your help guys! Let me know if I'm missing something important

 

[MarkFL]

Looks good to me!

 

[reardear]

thanks for your help! Would you mind showing me how I would do it using logs? I know I've probably did those types of questions using logs before but I don't recall how

 

[MarkFL]

You would only want to use logs if the variable which are solving for is in the exponent, i.e., you are trying to find out how long an investment must compound interest in order to reach a certain amount.

 

[JeffM]

thanks for your help! Would you mind showing me how I would do it using logs? I know I've probably did those types of questions using logs before but I don't recall how

You do not need to use logs, but you can if you want.

\(\displaystyle 38725 = 25000\left(1 + \dfrac{r}{26}\right)^{(20 * 26)} \implies log\left( \dfrac{38725}{25000}\right) = 520 * log\left(1 + \dfrac{r}{26}\right) \implies \dfrac{log(1.549)}{520} = log\left(1 + \dfrac{r}{26}\right) \implies\)

\(\displaystyle 0.000842 \approx log\left(1 + \dfrac{r}{26}\right) \implies antilog(0.000842) \approx 1 + \dfrac{r}{26} \implies 1.000842 - 1 \approx \dfrac{r}{26} \implies r \approx 26 * 0.000842 \implies r \approx 2.19\%.\)

It's the way we used to do it before calculators. Your way is the modern, quicker way.